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H
·
H
=
Definition 3.1.7
Let
be a real Hilbert space with the norm
.For
J
(
0
,T)
, the space
W
1
,p
(J
≤
≤∞
;
H
with
T>
0, and 1
p
)
is defined by
W
1
,p
(J
L
p
(J
u
∈
L
p
(J
;
H
)
:= {
u
∈
;
H
)
:
;
H
)
}
,
with the norm
(
J
p
H
+
p
H
u
(t)
d
t)
1
/p
u(t)
if 1
≤
p<
∞
,
u
W
1
,p
(J
;
H
)
:=
u
(t)
ess sup
J
(
u(t)
H
+
H
)
if
p
=∞
.
We again denote by
H
1
(J
W
1
,
2
(J
;
H
)
:=
;
H
)
.
3.2 Variational Parabolic Framework
Let
V
⊂
H
be Hilbert spaces with continuous, dense embedding. We identify
H
H
∗
with its dual
and obtain the triplet
V
⊂
H
≡
H
∗
⊂
V
∗
.
(3.6)
Denote by
(
·
,
·
)
the inner product on
H
, and let
·
V
,
·
H
be the norms on
V
H
L
2
(J
;
V
∗
)
and
and
H
, respectively. Furthermore, let
J
=
(
0
,T)
with
T>
0,
f
∈
u
0
∈
H
. Consider the variational setting of (
3.1
):
L
2
(J
H
1
(J
;
V
∗
)
such that
Find
u
∈
;
V
)
∩
d
d
t
u, v
V
∗
,
V
+
a(u,v)
=
f, v
V
∗
,
V
,
∀
v
∈
V
,
a.e. in
J,
(3.7)
u(
0
)
=
u
0
,
where
·
,
·
V
∗
,
V
denotes the extension of the
H
-inner product as
duality pairing
V
∗
×
V
in
. In particular, by Riesz representation theorem, we have
u, v
V
∗
,
V
=
(u, v)
, for all
u
∈
H
,
v
∈
V
.The
bilinear form a(
·
,
·
)
:
V
×
V
→ R
is associated
H
V
∗
)
in (
3.1
)via
with the operator
A
∈
L
(
V
,
a(u,v)
:= −
A
u, v
V
∗
,
V
,
∀
u, v
∈
V
,
where we denote by
L
(
V
,
W
)
the vector space of linear and continuous operators
A
:
V
→
W
.
Remark 3.2.1
d
d
t
in (
3.7
) is understood as a weak derivative.
(ii) The choice of the space
(i)
V
depends on the operator
A
. Often we can choose
L
2
. The choice of
H
=
V
is usually the closure of a dense subspace of smooth
functions, such as
C
0
, with respect to the 'energy' norm induced by
A
.
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